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Title: Classifying Hyperelliptic Trace Zero Varieties Susceptible To Low Genus Cover Attack
Author: Hosseini Lavasani, Seyed Ahmad
Advisor: Bauer, Mark
Keywords: Education--Mathematics
Issue Date: 24-Feb-2016
Abstract: A cover attack (on a Jacobian of a curve) is a method of decreasing the complexity of the discrete logarithm problem defined on the Jacobian of a curve by transferring it via a shared cover to the Jacobian of a new curve which admits a more efficient solution for this problem. We study different approaches that have been taken toward cover attacks and we propose new approaches for the problem of finding covers. We then concentrate on the case of cryptosystems based on the Trace Zero Variety (TZV) associated with hyperelliptic curves. We propose and implement an algorithmic solution to answer the question of finding the best cover to attack a given TZV of genus 2 curves. We extend current methods for constructing covers suitable for the attack. Subsequently, we construct all families of covers of genus 3, 4, and 5 which can potentially be used to attack such TZVs. In this way, we classify, as parametric families, all hyperelliptic curves which are vulnerable to a lower genus cover attack. As a result, we develop a method to avoid using hyperelliptic TZVs with a lower genus cover in cryptographic applications.
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